Can the Chain Rule be Applied to Show the Identity in Vector Calculus Homework?

[yy205001]

Homework Statement

\widetilde{F}(r)=F₁(r)i+F₂(r)j+F₃(r)k
\hat{r}=r/r
r(x,y,z)=xi+yj+zk, r=abs(r)=sqrt(x²+y²+z²)
(Hint: The chain rule will be helpful for this question.)

Show that:
\nabla\cdotF = \hat{r}\cdotdF/dr.

Homework Equations

The Attempt at a Solution

My attempt:
First, \nabla\cdotF=(dF₁/dr,dF₂/dr,dF₃/dr)

Then, Start on the RHS.
\hat{r}\cdotdF/dr
=\hat{r}\cdot(dF₁/dr,dF₂/dr,dF₃/dr)
=((x,y,z)/r)\cdot(dF₁/dr,dF₂/dr,dF₃/dr)
Now, i use the chain rule here.
=((x,y,z)/r)\cdot(dF₁/dx*dx/dr, dF₂/dy*dy/dr, dF₃/dz*dz/dr)

And i can't do further more here, can anyone help me on this?
Thanks in advanced!

 

[CompuChip]

The Attempt at a Solution

My attempt:
First, \nabla\cdotF=(dF₁/dr,dF₂/dr,dF₃/dr)

I'm pretty sure that's not how the gradient works.
First of all, the "definition" is
\vec\nabla = \left( \frac{d}{dx}, \frac{d}{dy}, \frac{d}{dz} \right)

Secondly, there is a dot in between, which means that you should get a scalar and not a vector like you have written.

 

[yy205001]

ops! So my definition is wrong! That's why i can't do it further more!
I can prove it now!
thanks CompuChip